In May of 2013, 84-year-old Gloria MacKenzie walked into a Publix supermarket in Zephyrhills, Florida, to buy a Powerball ticket.  Less than a month later, her son politely escorted her down to the lottery office to claim a record $590.5 million jackpot,1 completing what must be the best trip to a Publix supermarket in human history.

So how does a lottery jackpot get to over half-a-billion dollars?  Unlike most games in the U.S., which are run by state governments, Powerball is one of a handful of lotteries run by a consortium of states2, so players from all across the country pay into one giant kitty.  That can lead to some astronomical top prizes.  But is it worth playing?  I mean, if you're not Gloria MacKenzie?

First, the rules.  There are 59 white balls labeled {1, 2, ... , 59} and 35 red balls labeled {1, 2, ... , 35}.  Five of the white balls are drawn, along with just one of the red balls (the "Powerball" of eponymous fame).  A ticket costs $2, and consists of six numbers (five from 1 to 59, and a sixth from 1 to 35).  You win the jackpot if your ticket matches all five white balls, in any order, and also the Powerball.  So what's the probability that you win the grand prize?

Notice that there is only a single winning combination of numbers, so we only have to count the total number of draws possible.  A draw will consist of a combination of 5 balls chosen from a set of 59, along with a single ball chosen from a set of 35.  In other words, there are:

{ 59 \choose 5} {35 \choose 1} = 175, 223, 510

  possible draws, for a 1 in 175,223,510 chance of hitting it huge.  Not so awesome.

But there are other ways to win at Powerball besides matching every single number.  In fact, there are nine different payouts, depending on the number of matches, and whether your matches include the Powerball.  Here they are, along with their probabilities:

There are a couple of weird things going on here.  For one, notice that the payouts are wildly out of line with the probabilities.3  Also, different winning combinations might have the same payout, even though one combination might be twice as likely as the other.  Thirdly, why is the probability of matching just the Powerball not 1 in 35?4

That last question is actually helpful for figuring out the rest of the probabilities as well.  Sure, you have a 1 in 35 chance of matching the Powerball, but in order to qualify for the bottom prize, you have to also not match any of the white balls, otherwise you would get bumped up into a different prize category.  In other words, your ticket must match the Powerball, and include 5 of the 54 white balls not drawn.  A quick calculation will tell you that:

\frac{{54 \choose 5}{1 \choose 1}}{{59 \choose 5}{35 \choose 1}} \approx \frac{1}{54.41}

The rest of the probabilities can be verified using the same technique.  And once you have all of those, you can work out the expected value of a Powerball ticket.  Well, almost.  Since the jackpot varies from week to week, we can actually ask what the jackpot would need to be in order to produce an expected value of 0, which would represent the minimum grand prize required to justify buying a ticket.

It turns out the magic number is just under $280 million.  So, theoretically, any time the jackpot is greater than that, your Powerball purchase has a positive expected value.  Well, not quite.  So far we've ignored the fact that you're going to have to pay taxes on that money.  A lot of them.  And you're also probably better off, in terms of real dollars, taking the discounted lump sum and investing it rather than take the annuity.  In any case, your actual monetary gain is far, far less than the jackpot.  We've also ignored the probability of having to split the prize!  Jackpots get bigger, so more people buy tickets, so jackpots get bigger, etc.

The actual jackpot you would need to justify a ticket purchase, from the standpoint of expected value, is probably beyond record-setting.  Then again, you might walk into just the right Publix, on just the right day.  You just might make the paper.

Teachers, want to have this conversation in class?  Check out the lesson materials on our site!

1.  That's still (as of this writing) the largest Powerball jackpot in U.S. history, and #3 among all U.S. lottery games.  Unwilling to wait until her 114th birthday for the annuity to pay off, Gloria instead opted for the $370.9 million lump sum.  After taxes, she actually took home somewhere around $278 million.

2.  The Multi-State Lottery Association or — for somewhat impenetrable reasons — MUSL.

3.  And they are probabilities, even though they are (incorrectly) labeled as 'Odds.'

4. The MUSL gets this question so much that it appears on their Powerball FAQ page, which, if you have some time to kill, is hilarious and well written.  Hat tip.

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